Manuals/calci/TOEPLITZ

MATRIX("TOEPLITZ",order)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle order} is the size of the Toeplitz matrix.

Description

• This function gives the matrix of order 3 with the property of toeplitz matrix.
• A Toeplitz matrix is a matrix with the constant values along negative sloping diagonals(descending diagonal from left to right).
• If the i,j element of A is denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{i,j}} , then we have

$A_{i,j}=A_{i+1,j+1}=a_{i-j}$.

• Any nxn matrix A of the form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} \\ a_{1} & a_0 & a_{-1} & \cdots & \ddots & \vdots \\ a_{2} & a_{1} & \cdots& \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & a_{-1} & a_{-2}\\ \vdots & & \ddots & a_{1} & a_{0}& a_{-1} \\ a_{n-1} & \ldots & \ldots & a_{2} & a_{1} & a_{0} \end{bmatrix} }

• The property of Toeplitz matrix is :Toeplitz matrices are persymmetric.
• Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric.
• Toeplitz matrices commute asymptotically.

Examples

• MATRIX("toeplitz")

0.5852752963546664	0.5083035423886031	0.8240970941260457
0.5852752963546664	0.5852752963546664	0.5083035423886031
0.5083035423886031	0.5852752963546664	0.585275296354666

• MATRIX("toeplitz",5,1..7)

1	2	3	4	5
6	1	2	3	4
7	6	1	2	3
1	7	6	1	2
2	1	7	6	1

• MATRIX("toeplitz",4,761..770)

761	762	763	764
765	761	762	763
766	765	761	762
767	766	765	761

See Also

• PERSYMMETRIC
• PASCAL
• TRIANGULAR